• Optimal control is the problem of finding control strategies for a dynamic system such that a certain performance function is minimized (or maximized). The subject stems from the calculus of variations and was developed into an independent discipline during the early 1950s mainly due to two discoveries: the maximum principle by L.S. Pontryagin and dynamic programming by R. Bellman. Optimal control finds its application in a variety of areas including engineering, economics, biology, and logistics.
• In this course, we will explore both theoretical and numerical aspects of optimal control theory: mathematical formulation, necessary and sufficient conditions for optimality, and various methods and techniques for solving optimal control problems and approximating their solutions. Initially, students will be introduced to the main theoretical foundations of the theory, followed by the development of several applications through numerical methods.
• Prerequisites: basics of Functional Analysis, Differential Equations and PDEs; and Control Theory. Intended audience include engineering and computer science students with some PDEs background.

The following topics will be covered

• Introduction and Examples
• Calculus of Variations: Euler-Lagrange equations and necessary conditions for optimality
• Pontryagin's Maximum Principle and applications to linear and nonlinear systems
• Dynamic Programming: Bellman's Principle of Optimality Hamilton-Jacobi-Bellman (HJB) equation
• Linear Quadratic Regulator (LQR)
• Applications and Numerical Methods for Optimal Control.

Course slides and reading material will be available from the instructor.

• Optional Textbooks:
  • D. Liberzon. Calculus of variations and optimal control theory: A concise introduction, Princeton University Press 2011.
  • Donald E Kirk. Optimal control theory: An introduction. Courier Corporation 2012.
  • E. Trélat, Contrôle Optimal: Theorie et Applications, DE BOECK SUP, 2008.

1. Midterm Exam: 25%
2. Project: 25%
3. Final Exam: 50%